Approximation of stable random measures and applications to linear fractional stable integrals

TL;DR

This paper develops a lattice-based approximation method for stable processes represented by stochastic integrals, proving weak convergence and improving approximation schemes for linear fractional stable motions.

Contribution

It introduces a lattice approximation approach for stable processes and demonstrates weak convergence using a stable Lindeberg-Feller Theorem, enhancing existing methods for fractional stable integrals.

Findings

Lattice approximations converge weakly as mesh size decreases.

Improved approximation schemes for linear fractional stable motions.

Application of a stable Lindeberg-Feller Theorem to stochastic integrals.

Abstract

Using lattice approximations of Euclidean space, we develop a way to approximate stable processes that are represented by stochastic integrals over Euclidean space. Via a stable version of the Lindeberg-Feller Theorem we show that the approximations weakly converge as the mesh-size goes to zero. As an application, we improve upon previous approximation schemes for integrals with respect to linear fractional stable motions.